A Variant of D’alembert’s Matrix Functional Equation


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where <jats:italic>G</jats:italic> is a group that need not be abelian, and <jats:italic>σ</jats:italic> : <jats:italic>G → G</jats:italic> is an involutive automorphism of <jats:italic>G</jats:italic>. Our considerations are inspired by the papers [13, 14] in which the continuous solutions of the first equation on abelian topological groups were determined.</jats:p>


Introduction
Throughout this paper, let G be a group with neutral element e, and σ : G → G be a homomorphism such that σ • σ = id. Let M 2 (C) denote the algebra of complex 2 × 2 matrices. It will represent the range space of the solutions in this paper. The purpose of this paper is to solve the following matrix functional equation where Φ : G → M 2 (C) is the unknown function. The contribution of the present paper to the theory of matrix d'Alembert's functional equations lies in the study of (1.1) on groups that need not be abelian. On abelian groups the solutions of Eq. (1.1) are known: the matrix or even operator version (1.1) of d'Alembert's functional equation with σ = −id has for Φ(e) = I been treated by Fattorini ([7]), Kurepa ([9]), Baker and Davidson ( [2]), Kisyński ([8]), Székelyhidi ([17]), Chojnacki ([3]), Sinopoulos ([12]) under various conditions like G = 2G or the solution being bounded on G. For a general involutive automorphism σ not just σ = −id, Stetkaer ( [14]) determined the general solution Φ : G → M 2 (C) of (1.1). He did not need extra assumptions on the abelian topological group G and also found the solutions of (1.1) when Φ(e) = I.
The 2 × 2 matrix valued solutions of (1.2) and (1.3) are given in Corollaries 6.1 and 6.2, respectively. Example 5.5 shows that solutions of (1.2) are not generally abelian (see Notation). This is in contrast to the complex valued solutions of (1.2) which are multiplicative ( [15,Theorem 3.21]). We also show that any continuous solution of (1.1) on a compact group is abelian. Another main result of this paper is the solution of the functional equation the sine addition law and the symmetrized additive Cauchy equation play important roles in finding the solutions of the functional equation (1.1). The complex-valued solutions, where G is a semigroup, of (1.5), (1.8), and (1.9) were studied by Stetkaer in [16], [15,Chapter 4], and [15,Chapter 2], respectively, while the complex-valued solutions, where G is a possibly nonabelian group or monoid, of (1.6) and (1.7) were obtained by Fadli, Zeglami and Kabbaj in [5] and [6], respectively. General results about similar scalar functional equations on abelian groups are summarized in the monograph by Aczél and Dhombres [1] that contains many references. Notation. Throughout this paper we work in the following framework and with the following notation and terminology. We use it without explicit mentioning. G is a group that need not be abelian with neutral element e. Let id : G → G denote the identity map, and σ : G → G a homomorphism of G such that σ • σ = id. We let M 2 (C) the algebra of all complex 2 × 2 matrices, I its identity matrix and GL 2 (C) the group of its invertible matrices. We use the notation A(G) for the vector space of all additive maps from G to C, and put A ± (G) := {a ∈ A(G) : a • σ = ±a}.
By N (G, σ) we mean the set of the solutions θ : G → C of the homogeneous equation, namely Let S be a semigroup and X be a groupoid. A function f : S → X is multiplicative on S if f (xy) = f (x)f (y) for all x, y ∈ S. A character of G is a multiplicative function from G into C * . A function f : S → X is abelian, if for all x 1 , x 2 , · · · , x k ∈ S, all permutations π of k elements and all k = 2, 3, · · · . Any abelian function f is central, meaning f (xy) = f (yx) for all x, y ∈ S.

Auxiliary results
The following lemma presents some results that are essential for the proof of our first main result (Theorem 5.1).
Lemma 2.1. If the pair X, Z : G → C satisfies the functional equation where γ : G → C is a multiplicative function such that γ = 1, then X and Z are abelian functions.
Proof. For all x, y, z ∈ G we have Subtracting the two previous identities we get for all x, y, z ∈ G. Putting x = z in (2.2) we obtain Let z 0 ∈ G first satisfy that γ(z 0 ) = 1, then we get by (2.3) that Z(z 0 y) = Z(yz 0 ) for all y ∈ G.
Next, we show that X is abelian. Indeed, making the substitutions (x, yz) and (x, zy) in (2.1), we get respectively Subtracting the two previous identities, we get .
Changing x and y in (2.1) we see that the function X is central. Since X and Z are central functions, then X is abelian. From the equation (2.1) and since γ = 1 we can prove that Z is also abelian. Hence we get the claimed result.

A connection to the sine addition law
The following lemma lists pertinent basic properties of any solution Φ : G → M 2 (C) of (1.1) satisfying Φ(e) = I.
x ∈ G is also a solution of (1.1). Proof.
(2) Interchanging x and y in (1.1) we get and then replacing y by σ(y) in the last equation, we obtain by using (1) that (3) can be trivially shown.
Lemma 3.2 below derives an interesting connection between (1.1) and the sine addition matrix functional equation, viz.
Proof. Making the substitutions (ax, y), (σ(y)a, x) and (a, xy) in (1.1) we get respectively Subtracting the middle identity from the sum of the two others we get after some simplifications that shows that the functional equation (1.1) is connected with the sine addition matrix functional equation as follows:

Simultaneous triangularization
To set the stage let Φ : G → M 2 (C) be a solution of the functional equation (1.1), namely Suppose that Φ(e) = I. In view of Lemma 3.1 (2) the elements of the set {Φ(x), x ∈ G} commute pairwise. Then it is easy to verify after some computations that the elements of the following bigger set E = {Φ(x), Φ a (x) | x, a ∈ G} also commute pairwise, so by linear algebra all elements Φ(x), Φ a (x) of E can be brought into upper triangular form. Therefore there exist six functions φ 1 , φ 2 , ψ 1 , l 1,a , l 2,a , l 3,a : G → C, and a matrix P ∈ GL 2 (C) such that According to Lemma 3.1 the function x → C(x) = P −1 Φ(x)P, x ∈ G is also a solution of (1.1), so its components satisfy the following system of functional equations Likewise, the component functions of Φ a , a ∈ G satisfy the following system of equations By the definition of Φ a , the functions l 1,a , l 2,a and l 3,a can be expressed in terms of φ 1 , φ 2 and ψ 1 as follows: for all x ∈ G then the elements of the set {Φ(x) | x ∈ G} can be simultaneously diagonalized and so we may assume that ψ 1 = 0. Thus the system (4.2) becomes as follows: Otherwise we have φ := φ 1 = φ 2 and l 0,a := l 1,a = l 2,a where a ∈ G. Then by (4.2) combined with (4.3) we get: for all a, x, y ∈ G.

Main results
Putting x = y = e in (1.1) we get Φ(e) 2 = Φ(e), from which we see that Φ(e) : C 2 → C 2 is a projection, so there are only the following three cases: Φ(e) = 0, Φ(e) = I or Φ(e) is a 1-dimensional projection.
The first case implies that So from now on we are going to focus only on the other two cases. The first main theorem of the present paper concerns the second case: it highlights the form of the solutions Φ of the matrix functional equation (1.1) for which Φ(e) = I. It reads as follows: are the matrix valued functions of the three forms below in which P ranges over GL 2 (C): where χ 1 and χ 2 are characters of G. (2) where χ is a character of G such that χ = χ • σ and a ± ∈ A ± (G).
where χ is a character of G such that χ = χ • σ, ψ is a solution of the symmetrized additive Cauchy equation (1.9) such that ψ ∈ N (G, σ) and Proof. It is easy to verify with simple computations that all formulas above for Φ define solutions of (1.1). So it remains to show the other direction. So we assume that Φ : G → M 2 (C) is a solution of (1.1) such that Φ(e) = I. With the notation from Section 4, we have two cases: So we are in case (1) of our statement. Case 2: Suppose that φ 1 = φ 2 = φ, then for every a ∈ G we have l 1,a = l 2,a =: l 0,a . Since φ is a solution of (1.5), then from [16, Theorem 2.1] there exists a character χ of G such that Now, we are going to distinguish between two subcases: Subcase 2.1: If χ = χ • σ, then we get φ = χ. From (4.1) ψ 1 is a solution of the following equation: ψ 1 (xy) + ψ 1 (σ(y)x) = 2χ(x)ψ 1 (y) + 2ψ 1 (x)χ(y), x, y ∈ G.
Dividing (5.4) by χ(x)χ(y) and putting Γ := ψ 1 /χ, then we see that Γ is a solution of the variant of the quadratic functional equation which shows, according to [6,Theorem 5.4], that where B : G × G → C is a bi-additive function of G such that B(x, σ(y)) = −B(y, x) for all x, y ∈ G, and ψ is a solution of the symmetrized additive Cauchy equation (1.9) such that ψ ∈ N (G, σ). Hence we get So we are in case (3) of our statement. Subcase 2.2: Here χ = χ • σ. We will start by showing that Φ is abelian. According to (4.4) (l 0,a , φ), a ∈ G is a solution of the sine addition law, then from [15, Theorem 4.1] there exist α a ∈ C * such that Replacing φ and l 0,a into (4.4), then we get l 3,a (xy) = χ(x)H(y) + H(x)χ(y) + χ • σ(x)L(y) (5.5) Dividing (5.5) by χ(x)χ(y) gives us Similarly we can deduce easily that Z(e) = 0. Putting y = e in (5.6), we obtain So the functional equation (5.6) becomes where γ = 1, because χ = χ • σ. From Lemma 2.1 we get that X and Z are abelian. Then so are l 3,a = χX and L = χZ. From we infer that ψ 1 is also abelian. Therefore then the matrix Ω := C(x 2 0 ) − C(σ(x 0 )x 0 ) is invertible. Since the matrix 1 2 (C(x 2 0 ) − C(σ(x 0 )x 0 )) −1 is invertible, it has a square root K, which is a polynomial in Ω (see, e.g., [4, Chapter VII, Section 1]). Now C(x) commutes with Ω, so C(x) commutes with any polynomial in Ω, and in particular it commutes with K. Since C(x) for any x ∈ G is an upper triangular matrix, so is Ω. It follows that K, being a polynomial in Ω, is also upper triangular.
In a similar fashion as in the case of an abelian topological group ( [13]), we introduce another function, this time N , as which means that the function M is multiplicative. Moreover, using Lemma 3.1, we get Since the matrix-valued functions C(x), K and N (x), x ∈ G are upper triangular, where the diagonal elements of each function are equal, then by using the definition of M we may put M = m m 12 0 m . From (5.7) we get m + m • σ = χ + χ • σ, which implies by the linear independence of group homomorphisms from G into C * that m = χ or m = χ • σ.
As it is possible to exchange χ and χ • σ then we may assume that m = χ.
Since M is a multiplicative function, then we get Hence, a := m 12 /χ is an additive function. By using (5.7) we obtain which is equivalent to where a ± := a±a•σ 2 ∈ A ± (G). So we are in case (2) of our statement, which completes the proof.
Remark 5.2. If we assume that G is a topological group and that the function Φ : G → M 2 (C) is a continuous solution of (1.1) then the functions χ, χ 1 , χ 2 , a + , a − , S and ψ in Theorem 5.1 are continuous. Indeed, using [15,Theorem 3.18 (d)], it is easy to see that the characters in Theorem 5.1 are continuous. For the case (3) of Theorem 5.1, we have that g 1 := S + ψ by assumption is continuous. Hence so is g 2 (x) := g 1 (x 2 ), x ∈ G. But g 2 − 2g 1 = 2S, so S is continuous. ψ is also continuous, because ψ = g 1 − S. If we are in case (2) of Theorem 5.1, we can prove that a + and a − are continuous. In fact, we have x → N (x) = C(x 0 x) − C(σ(x 0 )x) and N • σ = −N are continuous. These yield that M = C + KN and M • σ = C • σ + N • σ = C − KN are continuous. Since a = m 12 /χ, we can deduce easily that a + and a − are continuous.
The second main theorem of the present paper concerns the third case: It describes the complete solutions Φ of (1.1) when Φ(e) is a 1-dimensional projection. It reads as follows: is a 1-dimensional projection, are the matrix valued functions of the two forms below in which P ∈ GL 2 (C): where χ is a character, β ∈ C and a − ∈ A − (G).
Proof. Let Φ : G → M 2 (C) be a solution of (1.1) such that Φ(e) is a 1dimensional projection. Then there exists P ∈ GL 2 (C) such that P −1 Φ(e)P = 1 0 0 0 . We write φ 1 φ 3 φ 2 φ 4 := P −1 ΦP. If we put y = e in (1.1), then we get that (5.10) From (5.10) it is easy to show that φ 3 = φ 4 = 0, so that we have Φ = Then simple computations show that φ 1 and φ 2 satisfy the following system of functional equations Thus from [5,Theorem 3.6] there exists a character χ of G such that where α, β ∈ C and a − ∈ A − (G). Since φ 2 (e) = 0, then we get And so we get the desired result. Conversely, it is easy to verify that any function Φ of the form (5.8) or (5.9) is a solution of (1.1) such that Φ(e) is a 1-dimensional projection.  where a and b range over C (see e.g., [15,Example 3.14]). We consider the functions of the form where a, b, c ∈ C, c = 0, and P ∈ GL 2 (C). It is elementary to check that these functions are non-abelian solutions of (1.1) on H 3 (R) in which σ = id because the complex-valued function is a solution of the symmetrized additive Cauchy equation (1.9) on H 3 (R) and is not even central (see [15,Example 12.4]).

Applications
By applying Theorems 5.1 and 5.3 we describe the matrix valued solutions of the symmetrized multiplicative Cauchy equation on groups.
Corollary 6.1. The non-zero solutions Φ : G → M 2 (C) of the matrix functional equation are the matrix valued functions of the three forms below in which P ranges over GL 2 (C): where χ 1 , χ 2 , χ are characters of G, and ψ is a solution of the symmetrized additive Cauchy equation (1.9).
Proof. The proof follows from Theorems 5.1 and 5.3.
As another application of our results we give, in the following corollary, a complete description of the solutions of the equation (1.3), that is where the unknown function takes its values in the complex 2 × 2 matrices. Setting x = y = e in (1.3), we get Φ 2 (e) = −Φ(e), which means that −Φ(e) (or equivalently I + Φ(e)) is a projection. (2) If Φ(e) = 0, then Φ has one of the following three forms below in which P ranges over GL 2 (C): where χ 1 and χ 2 are characters of G.
where χ is a character of G such that χ = χ • σ and a ± ∈ A ± (G).
where χ is a character of G such that χ = χ • σ, ψ is a solution of the symmetrized additive Cauchy equation (1.9)such that ψ ∈ N (G, σ) and S : G → C is a map of the form S(x) = B(x, x), x ∈ G, where B : G×G → C is a bi-additive function of G such that B(x, σ(y)) = −B(y, x).
(3) If I +Φ(e) is a 1-dimensional projection, then Φ has one of the two forms: where χ is a character of G, P ∈ GL 2 (C), β ∈ C and a − ∈ A − (G).
Proof. Let Φ : G → M 2 (C) be a solution of (1.3). If we add the identity matrix in the two sides of (1.3), we get that where Ψ := Φ+I. So, by applying Theorems 5.1 and 5.3 we obtain the claimed result.
Conversely, simple computations show that the above forms of Φ are solutions of (1.3). Now, we derive formulas for the continuous solutions of (1.1) on compact groups. Corollary 6.3. The non-zero continuous solutions Φ : G → M 2 (C) of (1.1), on a compact group, are the functions of the following two forms: where P ∈ GL 2 (C), χ, χ 1 , χ 2 are continuous characters of G and β ∈ C.
Proof. Let Φ : G → M 2 (C) be a non-zero continuous solution of (1.1) on a compact group. It is easy to see that the functions a − , χ in Theorem 5.3 are continuous and in view of Remark 5.2 the functions a + , a − , S, ψ and the characters in Theorem 5.1 are also continuous. Hence a + , a − and S are bounded because G is compact. So by [15,Exercise 2.5] we deduce that a ± ≡ 0.
We may use the same argument as in [15,Exercise 2.5] to show that S ≡ 0. From [15,Proposition 2.17] and [15,Corollary 12.6] we can prove that any continuous solution of (1.9) on a compact group will vanish. So the first direction deduces easily from Theorems 5.1 and 5.3.
Conversely, it is elementary to show that the above forms of Φ are solutions of (1.1).
Remark 6.4. Corollary 6.3 above implies that any continuous solution Φ : G → M 2 (C) of (1.1) on a compact group is abelian. Remark 6.5. On a compact group if Φ : G → M 2 (C) is a continuous solution of (6.1), then it is a multiplicative function. Example 5.5 shows that this result is not generally true in any group.

Solution of Eq. (1.4)
As another main result of this paper, we solve the matrix functional equation where M is a monoid, the function Φ to be determined takes its values in M 2 (C), and σ : M → M is a homomorphism such that σ • σ = id. Putting x = y = e in (7.1), we get that Φ(e) is nilpotent with index less than 2, then we have only the two possibilities: Φ(e) = 0 or Φ(e) is a nilpotent matrix with index 2.
In the following theorem we express the solutions of (7.1) in terms of the complex-valued solutions of the variant of the homogeneous equation, namely (7.2) θ(xy) − θ(σ(y)x) = 0, x, y ∈ M. where P ranges over GL 2 (C) and θ is a solution of (7.2).
Proof. It is easy to prove with simple computations that the above formula for Φ defines solutions of (7.1). So it remains to show the other direction. For that we are going to distinguish between two cases: Case 1: If Φ(e) = 0, then we can prove that each commutator of the form Φ(x)Φ(y) − Φ(y)Φ(x), x, y ∈ M is nilpotent. Indeed, by using Lemma 7.1 (2) and (3), we get (Φ(x)Φ(y) − Φ(y)Φ(x)) 2 = 0 for all x, y ∈ M.
From [11, Theorem 3.1] we get that φ 4 = 0 and so φ 3 is a solution of (7.2). Finally we have the desired form.
By the same procedure as in the proof of Theorem 7.2 we can prove the following result